LECTURE NOTES

# ELEMENTARY MATHEMATICS

## by William Chen and Xuan Duong

This set of notes has been compiled over a number of years. Chapters 1, 3 - 6 and 8 are based on lecture notes used by the second author in the second semester of 1998. Most of the remaining chapters were used for the first time in the second semester of 1999.

The material has been organized in such a way to create a single volume suitable for preparing the student for university mathematics.

The overall emphasis of this set of notes is on examples, of which there are many. The only way to study mathematics satisfactorily at this level is to study many examples and to attempt many exercises while going back regularly to the discussion in the notes for appropriate references.

To read the notes, click the links below for connection to the appropriate PDF files.

The material is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.

## SECTION A --- ALGEBRA

### Chapter 1 : BASIC ALGEBRA >>

• The Real Number System
• Arithmetic
• Distributive Laws
• Arithmetic of Fractions
• Factorization

### Chapter 2 : INTRODUCTION TO MATRICES >>

• Linear Equations
• Arithmetic
• Application to Linear Equations

### Chapter 3 : TRIGONOMETRY >>

• The Trigonometric Functions
• Some Trigonometric Identities

### Chapter 4 : INDICES AND LOGARITHMS >>

• Indices
• The Exponential Functions
• The Logarithmic Functions

### Chapter 5 : POLYNOMIAL EQUATIONS >>

• Linear Equations
• Factorization Again
• Higher Order Equations

### Chapter 6 : INEQUALITIES AND ABSOLUTE VALUES >>

• Some Simple Inequalities
• Absolute Values

### Chapter 7 : PROGRESSIONS >>

• Arithmetic Progressions
• Geometric Progressions

### Chapter 8 : ELEMENTARY COUNTING TECHNIQUES >>

• The Fundamental Principle of Counting
• Permutation
• Combination
• The Binomial Theorem
• Application to Probability Theory

### Chapter 9 : COMPLEX NUMBERS >>

• Introduction
• The Complex Plane

## SECTION B --- CALCULUS

### Chapter 10 : FUNCTIONS AND LINES >>

• Functions and Graphs
• Lines on the Plane

### Chapter 11 : INTRODUCTION TO DIFFERENTIATION >>

• Tangent to a Curve
• Arithmetic of Derivatives
• Derivatives of the Trigonometric Functions

### Chapter 12 : FURTHER TECHNIQUES OF DIFFERENTIATION >>

• The Chain Rule
• Implicit Differentiation
• Derivatives of the Exponential and Logarithmic Functions
• Derivatives of the Inverse Trigonometric Functions

### Chapter 13 : APPLICATIONS OF DIFFERENTIATION >>

• Second Derivatives
• Applications to Problem Solving
• Newton's Method

### Chapter 14 : INTRODUCTION TO INTEGRATION >>

• Antiderivatives
• Integration by Substitution
• Definite Integrals
• Areas